Photon-Pair Productionand Yang-Mills Equations

(1) Is it true that when a pair of particles is created from a Photon, that the Photon Energy (h x nu) is equal to 2(msub0)c^2 ?
If say an electron and positron are created does the photon Energy equal (at least) twice the mass of one electron 2(9.109 x 10^-31 kg)times c^2?

(2) Could anyone recommend the best Text Book that I should get if I want to study Yang-Mills Equations and Gauge Groups ? Is Group Theory in Real Analysis Textbooks?

Originally posted by joecoss (1) Is it true that when a pair of particles is created from a Photon, that the Photon Energy (h x nu) is equal to 2(msub0)c^2 ?
If say an electron and positron are created does the photon Energy equal (at least) twice the mass of one electron 2(9.109 x 10^-31 kg)times c^2?

Sorta! :)

You need to be a little bit careful here because of two important things:

1. To create a pair production from photons, the MINIMUM energy of the photon should be equal to the rest mass energy of the pair, exactly the way you calculated above. But note that I made sure you notice that this is the very minimum energy. This is because typically, if the photon just barely have enough energy to do this, this process is not very likely to occur.

When the photon does have more energy, this energy goes into the rest mass energy of the pair and whatever's leftover goes into their respective kinetic energy.

2. Notice one very glaring problem if you simply have a photon that just have enough energy to create the pair - violation of conservation of momentum. Before the pair production, you have a photon with energy hf, and momentum hbar*k. After the pair production, you have two particles with total rest energy equal to hf, but they just "sit" there with no kinetic energy and so zero net momentum overall. So energy here is conserved, but momentum isn't! What went wrong?

Simply based on this, pair production cannot occur in isolation. There has to be "something" else nearby to actually take up the recoil momentum of the photon. This means that a photon just doesn't make pair production in complete vacuum... What is typically done is that most pair production processes are obtained when we pass the photons through some crystal, and the crystal lattice is able to take up or supply the necessary momentum to preserve the conservation law.

Because of the energies involved (for electron-positron pair the minimum is 1.22 Mev), the recoil is by something much smaller than a lattice, for example an atomic nucleus in the electron-positron case.